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http -> https (the question was bumped anyway)
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Martin Sleziak
Martin Sleziak
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The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proofhere without proof in Wikipedia, and by induction it implies that for matrices $X_1,\dotsc, X_K$ and $p_1,\dotsc,p_K \in [1,\infty]$ with $\sum_i 1/p_i=1$, $$ \lvert\operatorname{Tr}(X_1\dotsm X_K)\rvert \leq \prod \lVert X_i\rVert_{p_i}.$$ In your setting, take $K=2N$, $X_{2k} = p^{a_k}$, $X_{2k-1} = A_k$, $p_{2k} = 1/a_k$ and $p_{2k-1} = \infty$.

I guess that all this is clearly explained (and with no mistake?) in Barry Simon's book Trace ideals and their applications.

The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proof in Wikipedia, and by induction it implies that for matrices $X_1,\dotsc, X_K$ and $p_1,\dotsc,p_K \in [1,\infty]$ with $\sum_i 1/p_i=1$, $$ \lvert\operatorname{Tr}(X_1\dotsm X_K)\rvert \leq \prod \lVert X_i\rVert_{p_i}.$$ In your setting, take $K=2N$, $X_{2k} = p^{a_k}$, $X_{2k-1} = A_k$, $p_{2k} = 1/a_k$ and $p_{2k-1} = \infty$.

I guess that all this is clearly explained (and with no mistake?) in Barry Simon's book Trace ideals and their applications.

The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proof in Wikipedia, and by induction it implies that for matrices $X_1,\dotsc, X_K$ and $p_1,\dotsc,p_K \in [1,\infty]$ with $\sum_i 1/p_i=1$, $$ \lvert\operatorname{Tr}(X_1\dotsm X_K)\rvert \leq \prod \lVert X_i\rVert_{p_i}.$$ In your setting, take $K=2N$, $X_{2k} = p^{a_k}$, $X_{2k-1} = A_k$, $p_{2k} = 1/a_k$ and $p_{2k-1} = \infty$.

I guess that all this is clearly explained (and with no mistake?) in Barry Simon's book Trace ideals and their applications.

Tidying, while this is on the front page
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LSpice
LSpice
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The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proof in wikipediaWikipedia, and by induction it implies that for matrices $X_1,\dots, X_K$$X_1,\dotsc, X_K$ and $p_1,\dots,p_K \in [1,\infty]$$p_1,\dotsc,p_K \in [1,\infty]$ with $\sum_i 1/p_i=1$, $$ |Tr(X_1\dots X_K)| \leq \prod \|X_i\|_{p_i}.$$$$ \lvert\operatorname{Tr}(X_1\dotsm X_K)\rvert \leq \prod \lVert X_i\rVert_{p_i}.$$ In your setting, take $K=2N$, $X_{2k} = p^{a_k}$, $X_{2k-1} = A_k$, $p_{2k} = 1/a_k$ and $p_{2k-1} = \infty$.

I guess that all this is clearly explained (and with no mistake?) in Barry Simon's book Trace ideals and their applicationsTrace ideals and their applications.

The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proof in wikipedia, and by induction it implies that for matrices $X_1,\dots, X_K$ and $p_1,\dots,p_K \in [1,\infty]$ with $\sum_i 1/p_i=1$, $$ |Tr(X_1\dots X_K)| \leq \prod \|X_i\|_{p_i}.$$ In your setting, take $K=2N$, $X_{2k} = p^{a_k}$, $X_{2k-1} = A_k$, $p_{2k} = 1/a_k$ and $p_{2k-1} = \infty$.

I guess that all this is clearly explained (and with no mistake?) in Barry Simon's book Trace ideals and their applications.

The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proof in Wikipedia, and by induction it implies that for matrices $X_1,\dotsc, X_K$ and $p_1,\dotsc,p_K \in [1,\infty]$ with $\sum_i 1/p_i=1$, $$ \lvert\operatorname{Tr}(X_1\dotsm X_K)\rvert \leq \prod \lVert X_i\rVert_{p_i}.$$ In your setting, take $K=2N$, $X_{2k} = p^{a_k}$, $X_{2k-1} = A_k$, $p_{2k} = 1/a_k$ and $p_{2k-1} = \infty$.

I guess that all this is clearly explained (and with no mistake?) in Barry Simon's book Trace ideals and their applications.

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Mikael de la Salle
Mikael de la Salle
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The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is here without proof in wikipedia, and by induction it implies that for matrices $X_1,\dots, X_K$ and $p_1,\dots,p_K \in [1,\infty]$ with $\sum_i 1/p_i=1$, $$ |Tr(X_1\dots X_K)| \leq \prod \|X_i\|_{p_i}.$$ In your setting, take $K=2N$, $X_{2k} = p^{a_k}$, $X_{2k-1} = A_k$, $p_{2k} = 1/a_k$ and $p_{2k-1} = \infty$.

I guess that all this is clearly explained (and with no mistake?) in Barry Simon's book Trace ideals and their applications.